To write the equations with the specified transformations of the parent function \( y = x^2 \) in the form \( y = a(x - h)^2 + k \), we can analyze each transformation:
A. Shifted left 1 and up 5
When we shift the graph to the left by 1, we replace \( x \) with \( x + 1 \). Shifting it up by 5 means adding 5 to the function.
Starting with the parent function: \[ y = x^2 \]
Transforming: \[ y = (x + 1)^2 + 5 \]
The final equation is: \[ \boxed{y = (x + 1)^2 + 5} \]
B. Reflected over the x-axis and stretched vertically by a factor of 3
Reflecting over the x-axis means we multiply the function by -1. Stretching vertically by a factor of 3 means multiplying the entire function by 3.
Starting with the parent function: \[ y = x^2 \]
Transforming: \[ y = -3(x^2) \] (Note: We can write this as \(y = -3(x - 0)^2 + 0\) for the specific form requested.)
The final equation is: \[ \boxed{y = -3(x - 0)^2 + 0} \]
C. Compressed vertically by a factor of \( \frac{1}{2} \) and shifted right 6
When compressing the graph vertically by a factor of \( \frac{1}{2} \), we multiply the function by \( \frac{1}{2} \). Shifting it right by 6 means replacing \( x \) with \( x - 6 \).
Starting with the parent function: \[ y = x^2 \]
Transforming: \[ y = \frac{1}{2}((x - 6)^2) \]
The final equation is: \[ \boxed{y = \frac{1}{2}(x - 6)^2} \]
In summary, the equations are:
A. \( y = (x + 1)^2 + 5 \)
B. \( y = -3(x - 0)^2 + 0 \)
C. \( y = \frac{1}{2}(x - 6)^2 \)
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